Lévy Flights in a Steep Potential Well

Abstract

Lévy flights in steeper than harmonic potentials have been shown to exhibit finite variance and a critical time at which a bifurcation from an initial mono-modal to a terminal bimodal distribution occurs (Chechkin et al., Phys. Rev. E 67, 010102(R) (2003)). In this paper, we present a detailed study of Lévy flights in potentials of the type U (x) ∝ |x| c with c > 2. Apart from the bifurcation into bimodality, we find the interesting result that for c > 4 a trimodal transient exists due to the temporal overlap between the decay of the central peak around the initial δ-condition and the building up of the two emerging side-peaks, which are characteristic for the stationary state. Thus, for certain system parameters there exists a transient tri-modal distribution of the Lévy flight. These properties of LFs in external potentials of the power-law type can be represented by certain phase diagrams. We also present details about the proof of multi-modality and the numerical procedures to establish the probability distribution of the process.

Lévy Flight Superdiffusion: An Introduction

“…From Refs. [Chechkin et al, 2004[Chechkin et al, , 2006. The main tools to investigate the barrier crossing problem for Lévy flights are the first passage times, crossing times, arrival time and residence times.…”

Transport in a Lévy ratchet: Group velocity and distribution spread

“…The interplay of deterministic dynamics and perturbative Lévy-type noises have been addressed in literature in various scenarios including several noise-induced effects like resonant activation [10,11], stochastic resonance [12], dynamical hysteresis [12,13], studies of decay/relaxation properties of the probability densities [14,15], escape from bounded intervals [16,17], the classical barrier crossing problem [18,19,20] or examination of stationary states [15,21,22,23]. However, very few examples [24] tackle the problem of Lévy noise driven dynamics of periodic systems.…”

Lattice-Boltzmann Simulation of Two-Dimensional Super-Diffusion

“…Koponen (1995) then proposed an exponentially tempered stable process, which was the focus of several recent studies (Baeumer and Meerschaert, 2010; Cartea and del-Castillo-Negrete, 2007; Rosinski, 2007;Zhang, 2010). The power-law truncation was also proposed by Chechkin et al (2008Chechkin et al ( , 2005Chechkin et al ( , 2004Chechkin et al ( and 2003 and Sokolov et al (2004). These studies show that the moments for particle jumps can converge if the probability for large jumps declines faster than the power-law behavior described by the standard stable distribution.…”